The function $f(x) =\log_e (\sin x), x ∈ (0,π)$ is

(A) strictly increasing on $(0,\frac{π}{2})$
(B) strictly decreasing on $(0,\frac{π}{2})$
(C) strictly increasing on $(\frac{π}{2},π)$
(D) strictly decreasing on $(\frac{π}{2},π)$
(E) strictly increasing on $(0,π)$

Choose the correct answer from the options given below.

(A) and (D) only

The correct answer is Option (1) → (A) and (D) only

$f(x)=\log_e(\sin x)$, domain $(0,\pi)$

$f'(x)=\frac{d}{dx}[\log_e(\sin x)]=\frac{\cos x}{\sin x}=\cot x$

For $(0,\frac{\pi}{2})$, $\cot x>0\Rightarrow f'(x)>0$ → $f(x)$ is increasing.

For $(\frac{\pi}{2},\pi)$, $\cot x<0\Rightarrow f'(x)<0$ → $f(x)$ is decreasing.

Correct statements:

(A) Strictly increasing on $(0,\frac{\pi}{2})$

(D) Strictly decreasing on $(\frac{\pi}{2},\pi)$